Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph
By Elizabeth Gillaspy and Jianchao Wu
Abstract
We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $\Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.
1. Introduction
The $C^*$-algebras associated to directed graphs Reference 5Reference 6Reference 9Reference 18Reference 19 have played an important role in $C^*$-algebra theory, largely because of the tight links between properties of the $C^*$-algebra, those of the underlying directed graph and those of the associated symbolic dynamical system. For instance, the $K$-theoryReference 8Reference 28 and the ideal structure Reference 1Reference 12 of graph $C^*$-algebras can be computed directly from the graph. The close structural ties between directed graphs and their $C^*$-algebras can also be used to identify $C^*$-algebras which are not graph $C^*$-algebras: for example, any simple $C^*$-algebra which is neither AF nor purely infinite — such as the $C^*$-algebras of noncommutative tori — cannot be a graph $C^*$-algebraReference 18.
Building on work of Robertson and Steger Reference 30, Kumjian and Pask introduced higher-rank graphs in Reference 17 to extend the successes of graph $C^*$-algebras to a broader class of $C^*$-algebras. The higher-rank graphs of rank $k$ (also called $k$-graphs) can be viewed as a $k$-dimensional generalization of directed graphs (which correspond to the case $k=1$), although they are formally defined as a countable category equipped with a degree functor. The construction of $k$-graph$C^*$-algebras generalizes that of graph $C^*$-algebras. As in the case of directed graphs, many structural properties of $k$-graph$C^*$-algebras are evident from the underlying $k$-graphs, such as their simplicity and ideal structure Reference 7Reference 16Reference 27Reference 29Reference 32, quasidiagonality Reference 4 and KMS states Reference 13Reference 14. Higher-rank graphs have also provided crucial examples Reference 2Reference 3Reference 25Reference 31Reference 33 for Elliott’s classification program for simple separable nuclear $C^*$-algebras.
Compared to the theory of graph $C^*$-algebras, a fascinating new feature of $k$-graphs with $k>1$ is the possibility to twist the construction of $C^*(\Lambda )$ with a 2-cocycle on $\Lambda$Reference 20, in a way that generalizes the construction of noncommutative tori. By expanding the class of higher-rank graph $C^*$-algebras to include twisted $k$-graph algebras, we vastly increase the class of $C^*$-algebras which we can analyze via the combinatorial perspective of higher-rank graphs. For example, the irrational rotation algebras arise as twisted $k$-graph algebras Reference 20, Example 7.7, but not as (untwisted) graph or higher-rank graph algebras Reference 10, Corollary 5.7.Footnote1
1
However, Reference 25, Example 6.5 shows that the irrational rotation algebras are Morita equivalent to certain 2-graph algebras.
Along with this extra flexibility comes a series of questions. What type of 2-cocycles are allowed? When do 2-cocycles induce the same twisted $C^*$-algebra? How do we compute with them? What is the relation between this construction and that of twisted groupoid $C^*$-algebras? In order to answer these questions, a systematic study of the cohomology groups of higher-rank graphs is in order.
Complicating the matter further is the fact that there is more than one construction of cohomology for a higher-rank graph. In view of the history of homological theories for topological spaces, categories, etc, this is not at all surprising — as the relation between singular and simplicial homology demonstrates, having multiple approaches can be a core strength of (co-)homology theories. For a higher-rank graph $\Lambda$ and a coefficient abelian group $M$, Kumjian, Pask, and Sims have defined both categorical cohomology groups $H^n_{} (\Lambda , M)$Reference 21 and cubical cohomology groups $H^n_{\operatorname {cub}} (\Lambda , M)$Reference 20. The latter can be viewed as the cohomology groups of the topological realization associated to a $k$-graph (Reference 15Reference 20) and lead to cocycles which are often easy to compute explicitly. The former are computed from composable tuples by treating a higher-rank graph as a small category; they are more flexible and make some theoretical results easier to obtain. A natural question was raised: are the cubical and categorical cohomology groups isomorphic?
In Reference 21, Kumjian, Pask, and Sims answered this question affirmatively in dimensions $n = 0$,$1$, and $2$. Furthermore, they provided explicit formulas for these isomorphisms on the cocycle level, making explicit computations possible. However, their proof methods were ad hoc and (in dimension 2) very technical. Although they conjectured that the cubical and categorical cohomology groups should agree in all dimensions, they suggested that a new approach would be needed.
We remark that establishing isomorphism in all dimensions is desirable, even if one is only interested in 2-cocycles. For one thing, a proof that works in full generality will likely be more natural and give us a better understanding of these cohomology groups. Perhaps more importantly, many crucial techniques in homological algebra involve long exact sequences — e.g., the long exact sequence associated to a change of coefficient groups — so an understanding of the entire collection of (co-)homology groups will be indispensable in applying these techniques to higher-rank graphs.
In this paper, we provide two proofs that the cubical and categorical (co-)homology groups for a higher-rank graph $\Lambda$ do indeed agree in all dimensions. The first proof is conceptual but abstract, while the second one is computational and provides explicit chain maps. As predicted by Kumjian, Pask, and Sims, our proof methods rely on a new (or at least unusual) approach to $k$-graphs: we view $k$-graphs primarily as categories. This contrasts with the usual combinatorial perspective on $k$-graphs, which views them as $k$-dimensional generalizations of directed graphs. We also make crucial use of the topological realizations of $k$-graphs, which were introduced in Reference 15. In addition to enhancing the (co-)homological tools available for the analysis of higher-rank graphs and their $C^*$-algebras, therefore, this paper demonstrates the utility of studying $k$-graphs from category-theoretic and topological perspectives. Indeed, the authors believe that the insight offered by these perspectives should shed more light on various problems involving twisted higher-rank graph $C^*$-algebras, such as their simplicity and $K$-theory; we plan to address these in future work.
This paper is organized as follows. We begin by reviewing the basics of higher-rank graphs, as well as some concepts from homological algebra, among which is the somewhat less common notion of a free $\Lambda$-module over a base (see Definition 2.6 or Reference 22), which will be of great use to us. In Section 3, we review the construction of cubical (co-)homology and generalize it to allow arbitrary modules as coefficients. We work in this generality throughout the paper.
Common to both of our proofs, in Section 4, we articulate our construction of a chain complex $\left({\mathbb{Z}\widetilde{Q}_n(\widetilde{\Lambda })}\right)_{n\in \mathbb{N}}$ of $\Lambda$-modules (see Construction 4.5), which we call the cubical free resolution associated to a $k$-graph, though the fact that it is a resolution (i.e., it is exact) is only made clear later. This chain complex computes the cubical (co-)homology groups via standard constructions (see Proposition 4.7). Connecting it with the categorical (co-)homology is thus the central issue for the rest of the paper, for which our two proofs diverge.
Our first proof of the isomorphism between the cubical and categorical (co-)homology groups of a $k$-graph$\Lambda$ builds on work of Kumjian and the first-named author Reference 11, which reinterprets the categorical cohomology groups of Reference 21 using the framework of modules over a small category. Standard arguments (cf. Reference 24, Corollary III.6.3) then imply that the categorical (co-)homology of $\Lambda$ can be computed by any free resolution of the trivial $\Lambda$-module$\mathbb{Z}^\Lambda$ — in particular, the cubical free resolution, provided that we can show it is indeed a resolution. This last point is the main goal of Section 5, which offers a proof of the exactness of $\left({\mathbb{Z}\widetilde{Q}_n(\widetilde{\Lambda })}\right)_{n\in \mathbb{N}}$ by showing that if a $k$-graph contains an initial object in the category-theoretic sense, then its topological realization is contractible (see Proposition 5.6). As a side comment, Remark 5.9 shows that whenever $\Lambda$ has an initial object, $C^*(\Lambda )$ is canonically isomorphic to the algebra of compact operators on the Hilbert space spanned by the vertices of $\Lambda$.
In Section 6, we complete our first proof (see Theorem 6.2) and discuss a few consequences. For example, our isomorphism implies that the categorical (co-)homology groups of a $k$-graph$\Lambda$ vanish in dimensions greater than $k$ and, at least when the coefficient is a constant module, only depend on the topological realization of $\Lambda$. These are not at all clear from the definition of categorical (co-)homology.
The remaining Section 7 details our more combinatorial second proof. In the same way as $\left({\mathbb{Z}\widetilde{Q}_n(\widetilde{\Lambda })}\right)_{n\in \mathbb{N}}$ computes the cubical (co-)homology, the categorical (co-)homology is defined in Reference 11Reference 21 by a chain complex $\left({\mathcal{P}_n}\right)_{n\in \mathbb{N}}$ of $\Lambda$-modules, which may be called the simplicial free resolution. Without using the knowledge that $\left({\mathbb{Z}\widetilde{Q}_n(\widetilde{\Lambda })}\right)_{n\in \mathbb{N}}$ is exact, we proceed by constructing explicit chain maps back and forth between $\left({\mathbb{Z}\widetilde{Q}_n(\widetilde{\Lambda })}\right)_{n\in \mathbb{N}}$ and $\left({\mathcal{P}_n}\right)_{n\in \mathbb{N}}$ that induce a chain homotopy equivalence between the two chain complexes (see Proposition 7.18); thus by standard homological algebra, these chain maps induce isomorphisms between the two types of (co-)homology groups (see Theorem 7.20). Intuitively speaking, we construct a “triangulation chain map” $\triangledown _*$ which “turns boxes into triangles” (that is, converts cubical $n$-chains into categorical $n$-chains), as well as a “cubulation chain map” $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_*$ which “turns triangles into boxes” (that is, converts categorical $n$-chains into cubical $n$-chains). It takes some nontrivial computations to verify these indeed form chain maps, i.e., they intertwine the boundary maps in the chain complexes (see Proposition 7.3 and Theorem 7.11). But once this is done, these chain maps can also be dualized to give cochain maps that induce isomorphisms between the cohomology groups, which enable us to convert cubical cocycles to categorical cocycles and vice versa.
The final parts of Section 7 prove the naturality of these (co-)chain maps (see Proposition 7.22) and compare them with the explicit isomorphisms of Reference 21 in degrees 0, 1, and 2. Up to a sign in degree 2, our isomorphisms agree with those of Reference 21.
2. Preliminaries
2.1. Higher-rank graphs
We begin by fixing some notational conventions. The natural numbers $\mathbb{N}$ will always include 0; we write $e_i$ for the canonical $i$-th generator of $\mathbb{N}^k$. If $n= (n_1, \ldots , n_k) \in \mathbb{N}^k$, we write $|n| \coloneq \sum _{i=1}^k n_i.$
We often view $\mathbb{N}^k$ as a small category with one object, namely 0, and with composition of morphisms given by addition. Thus, the notation $n \in \mathbb{N}^k$ means that $n$ is a morphism in the category $\mathbb{N}^k$. Inspired by this, we will follow the usual conventions for higher-rank graphs and use the arrows-only picture of category theory. That is, we identify the objects of a small category $\Lambda$ with its identity morphisms, and
Given $\lambda \in \Lambda$, we denote its source and range by $r(\lambda )$ and $s(\lambda )$ respectively. For $r \geq 1$, the collection of composable $r$-tuples in $\Lambda$ is
In this subsection, we recall some notions in homological algebra. We will generally follow the setting and terminologies of Reference 22, Section 9 and Reference 23, Section 2, whereby the notion of a free module over a base (Definition 2.6) is particularly useful for us.
Throughout this subsection $\Lambda$ denotes a small category. To be consistent with the notations for higher-rank graphs, we may use $v \Lambda w$ to denote the set of morphisms from $w$ to $v$, where $v,w \in \operatorname {Obj}\Lambda$.
Thus we may think of a $\Lambda$-module$\mathcal{M}$ as the single abelian group $|\mathcal{M}|$ instead of a family of abelian groups, which has the advantage of simplifying some notations.
The free $\Lambda$-modules which we now describe will play a central role in our arguments in this paper.
We now describe a few alternate characterizations of free $\Lambda$-modules. For the sake of simplicity, we restrict ourselves to right $\Lambda$-modules, though analogous statements can be made for left $\Lambda$-modules.
Next we give an intrinsic description of free modules, using the following “hom module” as a building block.
In order to work with (co-)homology, we also need the basics of (co-)chain complexes of $\Lambda$-modules.
2.3. Categorical (co-)homology
In this subsection, we review the construction of (co-)homology for small categories.
Thus a (co-)chain complex is acyclic if and only if all of its (co-)homology groups vanish. Two (co-)chain complexes that are homotopy equivalent have the same (co-)homology groups.
Remark 2.10 implies that any free resolution is also projective.
Any two projective resolutions of $\mathbb{Z}^\Lambda$ are homotopy equivalent (cf. Reference 24, Theorem III.6.1), making the following definition independent of the choice of a projective resolution.
3. Cubical (co-)homology with coefficients
In this section, we review the treatment of cubical (co-)homology for $k$-graphs, which was introduced by Kumjian, Pask, and Sims in Reference 20, before going on to explain how to incorporate general $\Lambda$-modules as coefficients. The main motivation for introducing cubical homology and cohomology is their ease for computation, as compared to categorical (co-)homology.
Throughout this subsection, $\Lambda$ denotes a $k$-graph.
More generally, mimicking the definitions in Remark 2.18, we can incorporate a coefficient left $\Lambda$-module into the definition of the cubical homology.
One readily verifies that $(\partial _n \otimes \operatorname {id}) \circ (\partial _{n+1} \otimes \operatorname {id}) = 0$ for all $n$.
Similarly, we can define the cubical cohomology with a right $\Lambda$-module as its coefficient.
One readily verifies that $\delta _{n+1} \circ \delta _{n} = 0$ for all $n$.
4. The cubical free resolution
In this section, we construct the cubical free resolution of a $k$-graph, which plays a central role in both of our proofs showing the isomorphism between cubical (co-)homology groups and categorical (co-)homology groups.
Next we consider cubical complexes over future path $k$-graphs.
As in the case of the categorical free resolution (Remark 2.16), in any generating tuple $(\eta , \lambda )$ of $\mathbb{Z}Q_n(\widetilde{\Lambda })(v)$, where $\eta \in Q_n(\Lambda )$ and $\lambda \in s(\eta ) \Lambda v$, only the second and final entry $\lambda$ is affected by the right $\Lambda$-module structure of $\mathbb{Z}Q_n(\widetilde{\Lambda })$. Intuitively speaking, the first entry $\eta$ carries the (cubical) homological information.
It follows that $\mathbb{Z}Q_n(\widetilde{\Lambda })$ is closely related to the cubical (co-)homology groups, which are defined respectively through the cubical chain complex $C_*^{\operatorname {cub}} ( \Lambda , \mathcal{N})$ in Construction 3.5 and the cubical cochain complex $C^*_{\operatorname {cub}} ( \Lambda , \mathcal{M})$ in Construction 3.7.
To justify the term “cubical free resolution”, we will need to show that the chain complex
of $\Lambda$-modules is acyclic. This will occupy more than half of the paper. We will provide two proofs of this fact, one topological and the other algebraic.
5. Initial vertices and contractibility
In this section, we show that if a $k$-graph has an initial vertex in the category-theoretical sense, then all of its reduced cubical homology groups vanish. This is a crucial step in our first proof of the isomorphism between cubical (co-)homology and categorical (co-)homology.
Let $\alpha$ be an initial vertex in a $k$-graph$\Lambda$. We evidently have $\alpha _\alpha = \alpha$. Moreover, for any $\lambda \in \Lambda$, we have $\alpha _{\alpha _\lambda } = \alpha _\lambda$ and $\alpha _\lambda = \alpha _{r(\lambda )} = \lambda \alpha _{s(\lambda )}$. This last equation implies that if $\lambda , \mu \in v \Lambda w$ we must have
and hence, by the factorization property, $\lambda = \mu$.
In order to study the reduced cubical homology groups of $k$-graphs with initial vertices, we invoke the notion of the topological realization $|\Lambda |$ of a $k$-graph$\Lambda$, introduced in Reference 15. It is a topological space whose homology groups coincide with the cubical homology groups of $\Lambda$ by Reference 20, Theorem 6.3.
In plain language, we associate to each morphism in $\Lambda$ a hyper-rectangle whose size equals $d(\lambda )$, and if two morphisms $\mu , \nu$ overlap on a hyper-rectangle $\lambda$, we glue the hyper-rectangles for $\mu$ and $\nu$ together along $\lambda$.
Although we will not use the following facts, we explore some further consequences of the existence of an initial vertex below.
6. Main results and consequences
We point out that this chain complex is merely a truncation of the complex $\mathbb{Z}\widetilde{Q}_*(\widetilde{\Lambda })$ introduced in Construction 4.5. We will refer to it as the cubical free resolution of $\Lambda$.
Next we detail a few of the consequences of the isomorphism $H^n_{\operatorname {cub}}(\Lambda , \mathcal{M}) \cong H^n(\Lambda , \mathcal{M})$ established in Theorem 6.2 above.
The next proposition uses the notation $M^\Lambda$ from Definition 2.3.
7. The chain maps
In this last section, we provide a second, algebraic, proof of the exactness of the cubical free resolution in Equation Equation 8, without using the topological constructions and results from Section 5. The advantage of our second approach is that it constructs explicit (co-)chain maps that implement the isomorphisms between the categorical and cubical (co-)homology groups, as these groups were originally defined in Reference 20. To be precise, this second proof does not rely on the result Reference 24, Theorem III.6.3 that allows one to use any projective resolution to compute the categorical (co-)homology. We anticipate that this approach may facilitate future computations. As examples of such computations, we establish naturality of our isomorphisms and compare our isomorphisms with those constructed in Reference 21 in degrees 0, 1, and 2.
These (co-)chain maps ultimately come from $\Lambda$-chain maps back and forth between the cubical free resolution
Viewing these as bi-infinite $\Lambda$-chain complexes, we keep the notations $\mathbb{Z}\widetilde{Q}_*(\widetilde{\Lambda })$ for the former and $\mathcal{P}_*(\Lambda )$ for the latter. We remind the reader that in both complexes, the right action of $\Lambda$ affects only the last component of a generating tuple; in other words, each generator’s pertinent homological information is carried in the previous entries in the tuple. Consequently, our $\Lambda$-chain maps will always leave the last entry in each tuple untouched.
7.1. Mapping cubes to composable tuples
We first fix some notation. The symmetric group on $\{1, \ldots , n\}$ is denoted by $\Sigma _{n}$, so that $\Sigma _{n-1}$ is a subgroup of $\Sigma _{n}$. For a permutation $\sigma \in \Sigma _{n}$, we write $\operatorname {sgn}(\sigma )$ for its sign, which takes value in $\{-1, 1\}$.
7.2. Mapping composable tuples to cubes
In this subsection, we construct a chain map $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_*\colon \mathcal{P}_*(\Lambda ) \to \mathbb{Z}\widetilde{Q}_*(\widetilde{\Lambda })$. To streamline the notation, we carry out computations of cubical chains in terms of general rectangles instead of just cubes. Let us specify how we do this.
Intuitively speaking, this lemma says $\lambda$ and $\mu$ can be glued along their common face, which is an $(n-1)$-dimensional hyperrectangle given by $\widehat{F}^1_{K,j}(\lambda ) = \widehat{F}^0_{K,j}(\mu )$, to form a larger $n$-dimensional hyperrectangle.
We are now in a position to define the maps $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_n$, for $n \in \mathbb{Z}$. These maps are described pictorially in the diagram of Remark 7.10, which may help the reader to follow the construction below.
We observe that $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_n = 0$ if $n > k$.
Our main task in this subsection is to prove the following:
For the proof, we will need to describe the elements in $\mathbb{Z}{\widetilde{Q}}_{n-1}( \widetilde{\Lambda })$ which arise when we compute $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_{n-1} \circ \partial _{n}^{\mathcal{P}} (\lambda _0, \ldots , \lambda _n)$ and $\partial _n \circ \mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_{n}(\lambda _0, \ldots , \lambda _n)$. These $(n-1)$-cubes, which we denote by $\widehat{\Xi }(\lambda _0, \ldots , \lambda _n; J, q)$, are constructed as follows.
In the following, Lemmas 7.14 and 7.16 respectively establish that, as claimed, the rectangles $\Xi (\lambda _0, \ldots , \lambda _n; J, q)$ appear in
The reader is encouraged to use the above diagrams to follow the proofs of these lemmas.
We now begin the proof of Theorem 7.11, that is, showing $\partial _n \circ \mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_n = \mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_{n-1} \circ \partial _n^{\mathcal{P}}$.
7.3. A proof of isomorphism by chain maps
We are ready to complete our second proof of the isomorphism between the cubical and categorical (co-)homology groups of a $k$-graph$\Lambda$.
The advantage of this alternative proof is that it provides explicit chain maps that induce isomorphisms between cubical and categorical (co-)homology groups. More precisely, the pair of $\Lambda$-chain maps $\triangledown _*$ and $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_*$ induce chain maps between categorical and cubical (co-)chain complexes (cf. Constructions 3.5 and 3.7, and Remark 2.18).
7.4. Naturality
We establish naturality of our chain maps. In the following, we use $\circ$ to denote a composition of two natural transformations, and juxtaposition to denote a composition of two functors or between a natural transformation and a functor.
This lemma leads to the naturality of the (co-)chain maps in Theorem 7.20. First we specify what type of naturality is considered.
Let $k \text{–} \mathfrak{graph} \text{–} \mathfrak{LMod}$ be the category such that its objects are pairs $(\Lambda , \mathcal{N})$, where $\Lambda$ is a $k$-graph and $\mathcal{N}$ is a left $\Lambda$-module, and a morphism from $(\Lambda , \mathcal{N})$ to $(\Lambda ', \mathcal{N}')$ is a pair $(\varphi , \psi )$, where $\varphi \colon \Lambda \to \Lambda '$ is a $k$-graph morphism and $\psi \colon \mathcal{N} \to \mathcal{N}' \varphi$ is a $\Lambda$-module map. Then $C_*^{} ( -, -)$ and $C_*^{\operatorname {cub}} (-,-)$ form functors from $k \text{–} \mathfrak{graph} \text{–} \mathfrak{LMod}$ to $\mathfrak{chain}$ upon defining
Similarly, let $k \text{–} \mathfrak{graph}^{\operatorname {op}} \text{–} \mathfrak{RMod}$ be the category such that its objects are pairs $(\Lambda , \mathcal{M})$, where $\Lambda$ is a $k$-graph and $\mathcal{M}$ is a right $\Lambda$-module, and a morphism from $(\Lambda , \mathcal{M})$ to $(\Lambda ', \mathcal{M}')$ is a pair $(\varphi , \psi )$, where $\varphi \colon \Lambda ' \to \Lambda$ is a $k$-graph morphism and $\psi \colon \mathcal{M} \varphi \to \mathcal{M}'$ is a $\Lambda '$-module map. Then $C^*_{} ( -, -)$ and $C^*_{\operatorname {cub}} (-,-)$ form functors from $k \text{–} \mathfrak{graph}^{\operatorname {op}} \text{–} \mathfrak{RMod}$ to $\mathfrak{cochain}$ upon defining
for any $n \in \mathbb{Z}$, any $f \in C^n_{} ( \Lambda , \mathcal{M})$, any $g \in C^n_{\operatorname {cub}} ( \Lambda , \mathcal{M})$, any $\left( \lambda _0, \ldots , \lambda _{n-1} \right) \in \Lambda ^{*n}$ and any $\lambda \in Q_n(\Lambda )$.
7.5. Cochain maps in low dimensions
In Reference 21, Kumjian, Pask, and Sims constructed explicit maps on the cocycles that induce isomorphisms between cubical and categorical cohomology groups in low dimensions ($n \leq 2$) and with constant coefficients. Fixing a constant right $\Lambda$-module$M^{\Lambda }$ associated to an abelian group $M$, we now compare our cochain maps $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}^{n}_{M^{\Lambda }}$ and $\triangledown ^{n}_{M^{\Lambda }}$ (see Theorem 7.20) with the maps defined in Reference 21, for $n = 0$,$1$, and $2$. For the sake of brevity, we write $M$ in place of $M^{\Lambda }$.
For $n=0$, identifying both 0-cubes and composable 0-tuples with vertices, we see that both $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_{0}$ and $\triangledown _{0}$ induce the identity map on vertices. Hence $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}^{0}_{M}$ and $\triangledown ^{0}_{M}$ give the identity map on $M$-valued functions over the vertices, which agrees with the construction in Reference 21, Remark 3.9.
When $n=1$, the recipe from Reference 21 for passing from a composable 1-tuple $\lambda \in \Lambda$ to a linear combination of 1-cubes can be found in Theorem 3.10 and in particular Equation (3.7) of Reference 21. To explain the recipe, we decompose an arbitrary $\lambda \in \Lambda$ as a product $\underline{\lambda }_k \underline{\lambda }_{k-1} \cdots \underline{\lambda }_1$, where $d(\underline{\lambda }_i) = d_i(\lambda ) e_{i}$ for all $i$, and then write $\underline{\lambda }_i = \underline{\lambda }_{i,1} \ldots \underline{\lambda }_{i,d_i(\lambda )}$ where $\underline{\lambda }_{i,j}$ is an edge of degree $e_i$ for all $i$ and $j$. Given a cubical 1-cocycle $f \in Z^1_{\operatorname {cub}} ( \Lambda , M) \subset C^1_{\operatorname {cub}} ( \Lambda , M)$, Kumjian, Pask, and Sims define a categorical 1-cocycle $\widetilde{f} \in Z^1_{} ( \Lambda , M)$ by
In fact, Reference 21, Theorem 3.10 establishes that $\widetilde{f} (\lambda ) = \sum _{i=1}^{r} f({\lambda }_{r})$ for any sequence $\lambda _1 , \ldots , \lambda _r$ of edges with $\lambda _1 \cdots \lambda _r = \lambda$. It is straightforward to check that $\widetilde{f} = \mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}^{1}_{M} (f)$ by identifying ${k \choose 1}$ with $\{ 1, \ldots , k\}$ and $\underline{\lambda }_{i,j}$ with $\lambda (b + (j -1) e_i , b + je_i)$, where $b$ is as in Theorem 7.20.
The aforementioned Reference 21, Theorem 3.10 also shows that the analogue of $\triangledown ^{1}_{M}$ for Kumjian, Pask, and Sims was induced by viewing each edge as a $1$-tuple in the obvious way; and indeed, if $\lambda$ is a 1-cube, then $\triangledown _1(\lambda , \mu ) = (\lambda , \mu )$.
In order to describe the analogue of $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}^{2}_{M}$ from Theorem 3.16 of Reference 21, we first recall some notation used in Reference 21. Namely, for $\lambda \in \Lambda$, we write
where $d(\lambda _i) = d_i(\lambda ) e_i$. With this notation, in Reference 21, Kumjian, Pask, and Sims associated to a composable 2-tuple $(\lambda , \mu )$ the collection of 2-cubes we need to “flip” in order to convert $(\overline{\lambda \mu })_1 (\overline{\lambda \mu })_2 \cdots (\overline{\lambda \mu })_{k}$ to $\overline{\lambda }_1 \overline{\lambda }_{2} \cdots \overline{\lambda }_k \overline{\mu }_1 \cdots \overline{\mu }_k$. The first such 2-cube is given by $\lambda \mu (d_1(\lambda )e_1 , (d_1(\lambda ) + d_1(\mu ) )e_1+ d_2(\lambda )e_2)$. Given a cubical $2$-cocycle$f \in Z^2_{\operatorname {cub}} ( \Lambda , M)$, they defined a categorical 2-cocycle $c_f$ such that $c_f(\lambda )$ is given by the sum of the values of $f$ on the aforementioned 2-cubes.
On the other hand, $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_2(\lambda , \mu , \nu )$ is the sum of the 2-cubes in $\widetilde{\Lambda }(s(\nu ))$ associated to $d_i(\lambda )e_i \times d_{j}(\mu )e_j$ for $j > i$. These are the 2-cubes we need to “flip” in order to convert $\left( \underline{\lambda \mu }_k \underline{\lambda \mu }_{k-1} \cdots \underline{\lambda \mu }_1 , \nu \right)$ to $\left( \underline{\lambda }_k \underline{\lambda }_{k-1} \cdots \underline{\lambda }_1 \underline{\mu }_k \cdots \underline{\mu _1} , \nu \right)$. In other words, to pass from Kumjian, Pask, and Sims’ procedure to $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_2$, we need to reverse our choice of ordering on the generators of $\mathbb{N}^k$.
It follows from the proof of Reference 21, Theorem 4.15 that reversing the color order takes a cubical 2-cocycle to its inverse. Thus, using $\mathbin{\mathchoice{\vcenter{\img[][7pt][7pt][{$\displaystyle\boxempty$}]{Images/img679b8aea3023cf1939b6bd92db57e293.svg}}}{\vcenter{\img[][7pt][7pt][{$\textstyle\boxempty$}]{Images/img8cab2c4caff2666910b6d93ea1a70ae1.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptstyle\boxempty$}]{Images/img04aa906d7a2cd26d830772d1ee3743d1.svg}}}{\vcenter{\img[][5pt][5pt][{$\scriptscriptstyle\boxempty$}]{Images/imgf93a6eecaf7f50661e7d125aeeba5392.svg}}}}_2$ instead of the procedure from Reference 21, Theorem 3.16 will remove the unfortunate minus sign which appears in the isomorphism of Reference 21, Theorem 4.15 between cubical and categorical 2-cohomology.
The same Theorem 4.15 establishes that Kumjian, Pask, and Sims used (a version of) the map $\triangledown _2$ to map 2-cubes to composable 2-tuples. More precisely, this follows from the third displayed equation in the proof of Reference 21, Theorem 4.15. Here, one sees that if $\lambda = \mu \nu = \nu ' \mu '$ where $d(\mu ) = d(\mu ') = e_i, d(\nu ) = d(\nu ') = e_j$ and $j > i$, then Kumjian, Pask, and Sims mapped the 2-cube $\lambda \in Q_2(\Lambda )$ to the element
Consequently, if we use the functor $\widetilde{\Lambda }: \Lambda \to k \text{–} \mathfrak{graph}$ to translate the Kumjian-Pask-Sims map into a map $\mathbb{Z}\widetilde{Q}_2(\widetilde{\Lambda }) \to \mathcal{P}_2(\Lambda )$, we obtain precisely the map $\triangledown _2$.
In terms of cocycles, then, a categorical 2-cocycle $f \in Z^2_{} ( \Lambda , M)$ induces a cubical 2-cocycle $\lambda \mapsto f(\mu , \nu ) - f(\nu ', \mu ')$. This is an analogue of the formula producing an alternating bicharacter from a 2-cocycle on the group $\mathbb{Z}^k$.
$$\begin{equation} J_{(i)} = \begin{cases} K_{(i)} \, , & 1 \leq i < l \\ K_{(i+1)} \, , & l \leq i \leq n-1 \end{cases} \; . \cssId{texmlid6}{\tag{21}} \end{equation}$$
Remark 7.15.
Lemma 7.16.
Remark 7.17.
Proposition 7.18.
Theorem 7.20.
Lemma 7.21.
Proposition 7.22.
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The first author was partially supported by the Deutsches Forschungsgemeinschaft via the SFB 878 “Groups, Geometry, and Actions.” The second author was partially supported by NSF grant #DMS–1564401.
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